Question: Maxim and Salma were asked to find an explicit formula for the sequence $54,63,72,81,...$, where the first term should be $f(1)$. Maxim said the formula is $f(n)=54+9n$. Salma said the formula is $f(n)=9+54n$. Which one of them is right? Choose 1 answer: Choose 1 answer: (Choice A) A Only Maxim (Choice B) B Only Salma (Choice C) C Both Maxim and Salma (Choice D) D Neither Maxim nor Salma
The general explicit formula for arithmetic sequences is ${a_1}+{d}(n-1)$, where ${a_1}$ is the first term and $ d$ is the common difference. The first term is ${54}$ and the common difference is ${9}$. ${+9\,\curvearrowright}$ ${+9\,\curvearrowright}$ ${+9\,\curvearrowright}$ ${54},$ $63,$ $72,$ $81,...$ We get the following formula. $f(n)={54}+{9}(n-1)$ We can now see that $f(n)=54+9n$ is not a correct formula, because the constant difference is added one extra time for each term. For instance, according to this formula, the value of the first term would be: $f(1)=54+9\cdot1=63$. However, according to our table of values, $f(1)=54$. So Maxim is definitely wrong. What about Salma? We can see that $f(n)=9+54n$ is also not a correct formula, because the constant difference in this formula is $54$, which is $6$ times the actual constant difference of $9$. So Salma is also wrong. Neither Maxim nor Salma got a correct explicit formula.